I understand that for any finite group $G$, there is an isomorphism from the group algebra $\mathbb{C}[G]$ to the direct sum of endomorphisms in which each irreducible representation of $G$ appears exactly once, i.e. $\bigoplus_V \text{End}(V)$.
But the set of notes that I am reading also says that the inverse of that isomoprhism can be given like this: to each $\phi$ in $\text{End}(V)$, it assigns the function taking $g$ to $\dfrac{\text{Tr}_V(\rho_V(g)\phi)}{\text{dim}(V)}$. The notes provide no explanation.
I am confused. Does this mean that in order to recover the $g$-value from $\Phi\in\bigoplus \text{End}(V)$, I take each component $\Phi_V$ and then compute the sum $\sum_V\dfrac{\text{Tr}_V(\rho_V(g)\Phi_V)}{\text{dim}(V)}$? And if so, how would this be proven?
Thanks!